At the rate of 9,000 punches per hour, how much total time is required to punch 1,500 cards, each card having 45 columns to be punched?

Introduction / Context:
This is a work and time problem that involves a constant rate of punching columns on cards. Each card has a fixed number of columns, and there is a known rate of punches per hour. By calculating the total number of punches needed and then dividing by the given rate, we can find how long the task will take.

Given Data / Assumptions:
- Number of cards to be punched = 1,500.
- Each card has 45 columns that must be punched.
- The punching machine operates at 9,000 punches per hour.
- The machine maintains this rate steadily throughout the task.

Concept / Approach:
The total work can be measured in number of punches required. The rate of work is given as punches per hour. Time can be found by the basic relation time = total work / rate. After computing the time in hours, we can convert the fractional part of an hour to minutes for a clearer answer.

Step-by-Step Solution:
Step 1: Compute the total number of punches needed. Step 2: Each card has 45 columns, so total punches = 1,500 * 45. Step 3: Multiply: 1,500 * 45 = 67,500 punches. Step 4: The punching rate is 9,000 punches per hour. Step 5: Use time = total work / rate, so time in hours = 67,500 / 9,000. Step 6: Simplify the fraction: 67,500 / 9,000 = 7.5 hours. Step 7: Convert 0.5 hours to minutes: 0.5 * 60 = 30 minutes. Step 8: Therefore, the required time is 7 hours and 30 minutes.

Verification / Alternative Check:
As a check, consider how many punches are done in 7.5 hours at 9,000 punches per hour. That is 9,000 * 7.5 = 67,500 punches, which matches exactly the required number of punches. Therefore, the time calculation is consistent. If you tried 8 hours instead, you would get 72,000 punches, which is more than necessary, showing that 8 hours is not the minimal correct answer.

Why Other Options Are Wrong:
Option A (8 hrs) corresponds to 72,000 punches, overshooting the requirement.
Option B (7 hrs 15 min) is 7.25 hours, giving 9,000 * 7.25 = 65,250 punches, which is short of 67,500 punches.
Option D (8 hrs 20 min) is even more than 8 hours and produces far more punches than needed.

Common Pitfalls:
A typical mistake is to miscalculate the total number of punches or to divide incorrectly, sometimes forgetting to convert fractional hours into minutes. Another error is to multiply the rate by the number of cards instead of number of columns, which ignores the structure of the problem.

Final Answer:
The time required to punch all the cards is 7 hrs 30 min.